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  • View in gallery

    (a) Locations of SWAN coarse, medium, and fine domains. (b) Fine domain bathymetry. White asterisk is the location of the wind measurements, and the AWAC location is denoted by A, along with other locations where wave energy spectra from SWAN were also analyzed.

  • View in gallery

    Observations of wave, tide and wind conditions over four diurnal sea-breeze cycles: (a) significant wave height, (b) peak wave period, (c) peak wave direction, (d) sea level recorded at the AWAC meter, (e) wind direction, and (f) wind speed recorded at the radar station. Also shown in red are the values from the RAMS model.

  • View in gallery

    The one-dimensional wave energy spectrum at 2-hourly intervals measured by the AWAC during the (a) growth and the (b) decay phase of the sea-breeze cycle on 19–20 Feb 2009 at Secret Harbour (115.73°E, 32.40°S). Frequency tails are f−5 (dashed) and f−3 (dashed–dotted).

  • View in gallery

    Nested model domains for SWAN, showing wind vectors from RAMS at onset of sea breeze (1300 LT 19 Feb). Coarse and medium plots show smaller domains and wind at every second RAMS grid point; fine plot shows wind at every RAMS grid point.

  • View in gallery

    SWAN one-dimensional wave energy spectra at location A at 2-hourly intervals during the (a) growth and the (b) decay phases of the sea-breeze cycle on 19–20 Feb 2009.

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    The root-mean-square error in the SWAN 1D spectra compared to AWAC (a) averaged in time as a function of frequency and (b) averaged in frequency as a function of time over 17–20 Feb for the Komen (Ko) and Alves–Banner (AB) SWAN whitecapping schemes. (c) Significant wave height for the two whitecapping schemes and for the AWAC (black).

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    Nondimensional peak frequency ν against nondimensional fetch χ at three offshore locations at hourly intervals between 0400 and 0700 LT 20 Feb. Also shown is the relationship according to Eq. (5) with accuracy limits from Young (1999).

  • View in gallery

    Wind field from RAMS (medium grid) for the afternoon of 18 Feb from the onset of the sea breeze at (a) 1400 LT then at (b) 1600 and (c) 1800 LT. Also shown are concentric lines of 20-, 40-, and 80-km radii, indicating distance from the AWAC site at Secret Harbour.

  • View in gallery

    At location A, two-dimensional (frequency, direction) spectra of (a) log(N) wave action density, (b) Sin and wind speed U10, (c) Sds, and (d) Snl. The local wind is indicated by the white arrow.

  • View in gallery

    2D spectra of (left) log(N), (middle) Sin, and (right) Snl every 4 h from 2000 LT 19 Feb to 1200 LT 20 Feb.

  • View in gallery

    As in Fig. 9, but at site E at 1200 LT 20 Feb.

  • View in gallery

    Two-dimensional energy spectrum at location A at 1200 LT 20 Feb for the SWAN run where wind is turned off at 2000 LT 19 Feb.

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Spectral Evolution of Nearshore Wave Energy during a Sea-Breeze Cycle

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  • 1 CSIRO Marine and Atmospheric Research, and CSIRO Wealth from Oceans Flagship, Floreat, Western Australia, Australia
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Abstract

From in situ measurements taken over several sea-breeze cycles off a beach in southwest (SW) Australia, the evolution of the one-dimensional spectrum of wave energy is observed to have a distinctive spectral shape. During the land-breeze phase of the cycle, lower rates of dissipation of wave energy are seen at high frequencies compared to midrange frequencies above the remnant wind-sea peak. A simulation of waves was performed using the Simulating Waves Nearshore (SWAN) model and produced the same spectral evolution, by generating longshore modes, as seen in the observations. The performance of whitecapping schemes available in SWAN was assessed, and the Alves–Banner scheme was found to best simulate the observed growth and decay of the wave spectra. During the onshore phase of the sea-breeze cycle, local wave growth is duration limited, and during the offshore land-breeze phase, wave growth is fetch limited. From an examination of the modeled two-dimensional spectra it is found that quadruplet interactions play a key role in spreading high-frequency wave energy in frequency and direction space.

Corresponding author address: Jim Gunson, CSIRO Marine and Atmospheric Research, Underwood Ave., Floreat WA 6014, Australia. E-mail: jim.gunson@csiro.au

Abstract

From in situ measurements taken over several sea-breeze cycles off a beach in southwest (SW) Australia, the evolution of the one-dimensional spectrum of wave energy is observed to have a distinctive spectral shape. During the land-breeze phase of the cycle, lower rates of dissipation of wave energy are seen at high frequencies compared to midrange frequencies above the remnant wind-sea peak. A simulation of waves was performed using the Simulating Waves Nearshore (SWAN) model and produced the same spectral evolution, by generating longshore modes, as seen in the observations. The performance of whitecapping schemes available in SWAN was assessed, and the Alves–Banner scheme was found to best simulate the observed growth and decay of the wave spectra. During the onshore phase of the sea-breeze cycle, local wave growth is duration limited, and during the offshore land-breeze phase, wave growth is fetch limited. From an examination of the modeled two-dimensional spectra it is found that quadruplet interactions play a key role in spreading high-frequency wave energy in frequency and direction space.

Corresponding author address: Jim Gunson, CSIRO Marine and Atmospheric Research, Underwood Ave., Floreat WA 6014, Australia. E-mail: jim.gunson@csiro.au

1. Introduction

The diurnal sea-breeze and land-breeze cycle, caused by differential heating between land and sea, is a common feature in atmospheric circulation in tropical and subtropical coastal regions (Abbs and Physick 1992). In southwest Western Australia, the intensity of the sea breeze is also linked to the presence of a north–south low pressure trough located inland of the coast (Masselink and Pattiaratchi 2001), producing a southeasterly synoptic wind pattern. As the sea breeze develops, the wind veers toward the south-southwest, producing a locally generated wind sea oblique to the approximately north–south coastline (Lemm et al. 1999; Masselink and Pattiaratchi 2001). During the onshore phase of the sea breeze, the significant wave height of the locally generated wind-sea increases, with a peak frequency that appears at around 0.35 Hz and gradually moves to lower frequencies around 0.15 Hz near the peak of the sea breeze (Pattiaratchi et al. 1997; Masselink and Pattiaratchi 2001; Neetu et al. 2006). The increase in wave height also leads to an increase in wave-driven currents in the near shore (Pattiaratchi et al. 1997) and has been linked to the removal of beach cusps formed during the land-breeze phase (Masselink and Pattiaratchi 1998).

Wave growth due to the increasing wind is relatively well understood since the Joint North Sea Wave Project (JONSWAP) experiment (Hasselmann et al. 1973), and the JONSWAP wave energy spectrum for deep-water wind-generated waves has a shape that has been well established. Extensive descriptions of the current state of understanding of the processes governing spectra wave evolution may be found in Tolman et al. (2013) and Cavaleri et al. (2007). In particular, under a steady wind a fetch-limited wind sea has a well-defined spectral shape above the peak frequency. In deep water, the JONSWAP spectrum has a high-frequency tail E(f) ~ f−5, and in shallow water this shape slowly evolves to f−3 along the fetch (Holthuijsen 2007). The spectral shape is maintained by nonlinear transfer by nonlinear four-wave interactions (hereinafter quadruplet interactions) and whitecapping.

Although whitecapping has been extensively studied (Ardhuin et al. 2010; Cavaleri et al. 2007; Donelan and Yuan 1994), it is one of the least understood processes that are parameterized in spectral wave models. The third-generation spectral wave model Simulating Waves Nearshore (SWAN) has been used in a number of studies of dissipation through whitecapping (e.g., Rogers et al. 2003). Young and Verhagen (1996a,b,c), from measurements in a long lake, examined the role that quadruplet interactions play in determining the frequency and directional spectra of wind-generated waves in shallow water.

There has been less study of the spectral response to changes in both wind speed and direction. During a typical sea-breeze cycle, wind speed and direction change rapidly, and the opportunity exists to study the growth and decay of the local wind sea over time scales of a few hours. During the onshore phase of the sea-breeze cycle, typically beginning late morning to early afternoon, the wind speed builds, rapidly increasing from 5 to 15 m s−1 over a few hours. The local wind-sea growth has no fetch limit in the upwind direction and is thus termed duration limited. At night the wind speed drops and the wind direction veers offshore in the early hours of the morning. The local wind-sea growth is sensitive to the distance upwind to the shoreline and is thus termed fetch limited.

Young and van Agthoven (1997), using a duration-limited spectral model with a full solution of the nonlinear term, showed how following an increase in wind speed the nonlinear interactions act to stabilize the frequency spectrum. For changes in wind directions, Young et al. (1987) showed how for sudden wind shifts of less than 90°, the turning wind-sea peak was smoothed out rather than creating a secondary peak.

Ardhuin et al. (2007), using observations from the North Carolina–Virginia shelf [Shoaling Waves Experiment (SHOWEX)] and Duck, North Carolina (SandyDuck), and output from the WAVEWATCH III model, studied the spectral response of wind-sea growth in a slanting-fetch case, which occurs when the wind blows obliquely off a coast. In such cases, significant wave energy is found up to 75° from the wind direction with low-frequency components propagating alongshore and higher-frequency components, above the peak frequency, aligned with the wind. Bottema and van Vledder (2008) modeled slanting fetch using SWAN and found sensitivity to the numerical computation of the nonlinear wave–wave interactions and also found that the mean wave direction can deviate from the mean wind direction by 30°.

In this study, we examine the evolution of wave spectra in response to rapid changes in wind speed and direction during a diurnal sea-breeze cycle. The Regional Atmospheric Modeling System (RAMS) model is used to simulate the spatial and temporal variability of the wind field associated with the local sea breeze and larger-scale synoptic conditions. The SWAN wave model was implemented to simulate wave conditions during several sea-breeze/land-breeze cycles. Wave energy dissipation is an important component of the model, and several parameterizations of whitecapping were investigated. A fetch analysis shows both duration-limited and fetch-limited situations. Finally, analysis of the modeled two-dimensional spectra at various locations reveals the processes at play in SWAN affecting the spectral and directional distribution of wave energy near the coast.

2. Observations

The field site was at Secret Harbour, a relatively longshore uniform beach in southwest Australia (see Fig. 1), and the observation period was 10–28 February 2009. The region experiences mixed microtidal tides, and during the course of the observations reported below were dominated by diurnal variations with a mean spring range of 0.9 m. The observations presented here were part of a larger study to measure the nearshore response of waves and currents to sea-breeze forcing (Contardo and Symonds 2013). An acoustic wave and current profiler (AWAC) was situated 1000 m offshore (hereinafter denoted location A, shown in Fig. 1) in 8.9-m water depth.

Fig. 1.
Fig. 1.

(a) Locations of SWAN coarse, medium, and fine domains. (b) Fine domain bathymetry. White asterisk is the location of the wind measurements, and the AWAC location is denoted by A, along with other locations where wave energy spectra from SWAN were also analyzed.

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

The bottom-mounted AWAC measures the sea surface elevation at 2 Hz, using acoustic surface tracking from one vertical beam, and the near-surface orbital velocity is calculated from 1-Hz data from three slanted beams. Hourly bursts of sea surface elevation and velocity collected 4096 and 2048 data points, respectively. The time series of sea surface elevation were cosine tapered and one-dimensional spectra were computed via a fast Fourier transform (FFT) and smoothed in frequency with a 64-point Hanning window, giving 237 degrees of freedom. The two-dimensional spectra were calculated from the sea surface elevation and near-surface orbital velocity time series using the maximum likelihood method, a means of estimating directional wave spectra from spatial arrays of wave measurements (Hauser et al. 2005). Integrated parameters such as significant wave height (Hm0), peak frequency (Tp), and peak direction (Dir) are computed from these spectra.

A line of sand dunes, running parallel to the shoreline, is situated about 20 m away from the shoreline at Secret Harbour. Wind measurements were obtained at the top of the dunes about 13 m above mean sea level (location of white asterisk in Fig. 1) with 1-min averages of wind speed and direction measured every 10 min for the duration of the field program. Three consecutive sea-breeze cycles were observed on 17, 18, and 19 February as shown in the time series of wind speed and direction in Figs. 2e and 2f, respectively. During the peak of the sea breeze, the maximum wind speed is between 10 and 15 m s−1 from the southwest. For the corresponding time series of Hm0, the peak wave period and direction are shown in Figs. 2a, 2b, and 2c. During the peak of the sea breeze, Hm0 increases by about a factor of 2, the peak period decreases from 12 to 15 s to less than 5 s, and the wave direction veers toward the south. These values are typical for the summer wave conditions near Perth (Lemm et al. 1999). The corresponding tidal elevations recorded at the AWAC are shown in Fig. 2d.

Fig. 2.
Fig. 2.

Observations of wave, tide and wind conditions over four diurnal sea-breeze cycles: (a) significant wave height, (b) peak wave period, (c) peak wave direction, (d) sea level recorded at the AWAC meter, (e) wind direction, and (f) wind speed recorded at the radar station. Also shown in red are the values from the RAMS model.

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

The observed 1D spectra on 19–20 February are shown in Fig. 3a and show a persistent swell peak at 0.08 Hz with a second peak at higher frequencies associated with the locally generated wind sea. During the afternoon the peak frequency of the wind sea decreases from 0.4 to 0.25 Hz, and the energy increases, reaching a maximum at 1800. These are typical wave conditions for the summer sea breeze near Perth, generally a swell-dominated system, changing to a wind sea–dominated system during the peak of the sea breeze (Pattiaratchi et al.1997). These 1D spectra show similar evolution to those observed by Masselink and Pattiaratchi (2001) at two wave buoys: one offshore in 48-m water depth and one inshore in 17-m water depth, both approximately 40 km to the north of Secret Harbour. The dashed and dashed–dotted lines in Fig. 3 show the f−5 and f3 high-frequency tails. One can see that the growing wind sea lies between these two curves at midfrequencies, consistent with previous studies (Holthuijsen 2007).

Fig. 3.
Fig. 3.

The one-dimensional wave energy spectrum at 2-hourly intervals measured by the AWAC during the (a) growth and the (b) decay phase of the sea-breeze cycle on 19–20 Feb 2009 at Secret Harbour (115.73°E, 32.40°S). Frequency tails are f−5 (dashed) and f−3 (dashed–dotted).

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

From about 2000 LT onward, the wind sea decays as the wind speed decreases and veers through south to east. During the decay phase, as wave energy decreased at all frequencies above the swell peak, there was a slower decay of wave energy at f > 0.5 Hz, as can be seen in Fig. 3b. A spectral trough at 0.4–0.5 Hz appears at about 0010 LT and deepens over the next 6 h, while wave energy at higher frequencies does not decrease significantly, from the high-frequency tail, over that time. This apparent persistence of high energy at high frequencies, along with the apparent decay at midfrequencies, motivated the modeling analysis that follows, and this observed phenomena will be discussed in section 5a.

3. Modeling winds

In southwest Western Australia the strongest sea breezes often occur when a synoptic low pressure trough lies slightly inland along the coast, directing a southeast flow over coastal waters (Pattiaratchi et al. 1997; Masselink and Pattiaratchi 2001). To simulate the spatial and temporal variability of the complete sea-breeze cell, a series of nested model runs using RAMS (Pielke et al. 1992) was used. Figure 1 shows the three model domains, the largest domain having 16-km resolution, the medium domain having 4 km, and the smallest domain having 1 km. An example of the model winds is shown in Fig. 4 at the onset of the sea breeze on 19 February over all the domains. The modeled winds capture the sea-breeze cell, which extends out to approximately 115°E, and also contain the larger-scale synoptic wind patterns. The south to southeast synoptic wind pattern can be seen offshore, while closer to the coast the wind veers toward the southwest because of the sea breeze. These model results show the same spatial character, at synoptic scales, as the summer sea-breeze conditions described in Masselink and Pattiaratchi (2001). The time series of model wind direction compared well with observed values at the experiment site (Fig. 2e). Modeled wind speed shows some difference with observed values (Fig. 2f), with a noticeable overestimation during land-breeze phases. This discrepancy is probably because of the insufficient topographic resolution over the coast in the model (D. Abs 2009, personal communication; RAMS modeler), which introduces some error into the modeled wave growth during the land breeze

Fig. 4.
Fig. 4.

Nested model domains for SWAN, showing wind vectors from RAMS at onset of sea breeze (1300 LT 19 Feb). Coarse and medium plots show smaller domains and wind at every second RAMS grid point; fine plot shows wind at every RAMS grid point.

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

.

4. Modeling waves

SWAN is a spectral wave model (Booij et al. 1999; Ris et al. 1999), which evolves the wave action density N(σ, θ;x, y, t) (defined as wave energy density divided by intrinsic frequency E/σ) in frequency, direction, space, and time according to the action balance equation:
e1
where cg,x, cg,y are the propagation velocities in x and y space, respectively; cσ and cθ are the propagation velocities in σ and θ space, respectively; and S ≈ Sin + Snl + Sds is the sum of source/sink terms. The quantity Sin(f, θ) represents wind generation of wave energy, and Snl(f, θ) represents nonlinear wave–wave interactions; that is, both quadruplet and triad interactions (the latter were found to be negligible) and Sds(f, θ) represents dissipation through whitecapping, bottom friction, and depth-induced breaking. For the high-frequency wave energy that will be considered here, it may be assumed that Sds is dominated by whitecapping. The details of the formulation of the source/sink terms used in SWAN may be found in Holthuijsen (2007).
SWAN was run in third-generation mode using the default wind input term given by
e2
(where the linear wave growth term is turned off). The coefficient for exponential growth is of the form (Snyder et al.1981; Komen et al. 1984)
e3
where is the friction velocity , and c is the phase speed. Note that for shallow-water waves, . Hence, the size of the wind input term may be sensitive to the depth as well as the wind speed. Both this scheme and the Janssen (1991) scheme were tested in the SWAN runs.

Also of importance for later analysis, the quadruplet interactions term Snl is calculated in SWAN using the discrete interaction approximation (DIA; Hasselmann et al. 1985), which is a computationally effective approximation to an exact calculation of the nonlinear interactions. The DIA scheme is known to produce larger directional spreading than the exact calculation (Bottema and van Vledder 2008; Ardhuin et al. 2007). The DIA scheme was also found to produce weaker wave steering (difference between mean wave direction and wind direction) in slanting-fetch conditions (Bottema and van Vledder 2008) and to produce a broader wind-sea peak in frequency space with overpredicted wave energy at frequencies below the peak (Rogers and van Vledder 2013).

SWAN, at version 40.81, was run over the region in nonstationary mode with a time step of 15 min, forced by hourly winds from RAMS. Three nested wave model meshes (at 2000-, 400-, and 30-m grid resolution) were implemented replicating the RAMS domains shown in Fig. 4. In spectral space, 32 frequencies were used, spaced geometrically between 0.0464 and 1 Hz, with a directional resolution of 10°.

The bathymetry for the fine grid (shown in Fig. 1) was taken from a laser airborne depth survey (Department of Transport 2009) merged with Geoscience Australia bathymetry farther offshore. A line of fringing limestone reefs is apparent several kilometers offshore, which can dissipate incoming wave energy (Masselink and Pattiaratchi 2001). Tidal elevations were not used in the simulations, as they are quite small. The swell peak seen in Fig. 3 is from distant wave energy heading in an east-northeast (ENE) direction as observed by the AWAC. An estimate of the swell parameters [Hm0 = 2 m, Tp = 13 s, Dir = 40° (Cartesian)] was added as a boundary condition to the coarsest model run.

5. Results

Using the RAMS wind fields, SWAN was used to simulate the wave growth and decay through the sea-breeze cycle. Figure 5 shows the modeled 1D spectra at the same location and times as in Fig. 3. The left column shows the modeled spectra during the growth phase of the sea breeze, while the right column shows model spectra from the decay phase.

Fig. 5.
Fig. 5.

SWAN one-dimensional wave energy spectra at location A at 2-hourly intervals during the (a) growth and the (b) decay phases of the sea-breeze cycle on 19–20 Feb 2009.

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

The modeled 1D spectra display similar spectral evolution, in particular the enhanced decay of wave energy at midfrequencies, as seen in the observations shown in Fig. 2. These results were obtained using the wind input term based on Snyder et al. (1981). Corresponding runs using the more widely accepted scheme proposed by Janssen (1991) produced similar spectra but without the enhanced decay at midfrequencies. It is not obvious as to why the schemes produce different wave energy levels at midfrequencies. For this particular nearshore setting during a sea-breeze/land-breeze cycle, the Snyder wind input term produces a better simulation of the observed spectra.

The modeled 1D spectra were also found to be sensitive to the choice of whitecapping scheme used in SWAN. The default scheme follows the formulation proposed by Komen et al. (1984), hereinafter referred to as Ko. An alternative whitecapping scheme is based on the work of Alves and Banner (2003) and implemented in SWAN following van der Westhuysen et al. (2007), hereinafter referred to as AB. The average root-mean-square error between the 1D spectra from the AWAC and SWAN for 17–20 February is shown in Fig. 6 for the AB and the default Ko white-capping formulations. In Fig. 6a, the root-mean-square error at each frequency is time averaged for all spectra from 1000 LT on 19 February to 1000 LT on 20 February. The largest errors are found in the vicinity of the peaks in swell energy and wind-sea energy. In Fig. 6b, the root-mean-square error is frequency averaged for each spectrum and plotted as a function of time. The largest errors occur around the times of the peaks of the sea breeze. Comparison of the evolution of modeled and observed Hm0 over this period (Fig. 6c) show that Ko consistently overestimates Hm0 compared to AB. Overall, the AB white-capping formulation produced smaller errors, and this scheme was employed for all the SWAN results in the following sections.

Fig. 6.
Fig. 6.

The root-mean-square error in the SWAN 1D spectra compared to AWAC (a) averaged in time as a function of frequency and (b) averaged in frequency as a function of time over 17–20 Feb for the Komen (Ko) and Alves–Banner (AB) SWAN whitecapping schemes. (c) Significant wave height for the two whitecapping schemes and for the AWAC (black).

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

a. Wave growth

We now investigate how closely wind seas generated by a sea breeze may be represented by idealized fetch-limited growth approximations. Following Young (1999), we introduce the following parameters: nondimensional peak frequency and nondimensional fetch , where x is the fetch and is the peak frequency of the wind sea. The latter quantity is calculated from the one-dimensional energy spectra E by (Young 1999; Young and Verhagen 1996a)
e4
where the spectrum is integrated from 0.15 to 1.0 Hz to remove the swell energy.
There have been many studies on the dependence of wave energy and peak frequency on fetch (Babanin and Soloviev 1998; Dobson et al. 1989; Kahma and Calkoen 1992). From examining a number of previous studies of fetch-limited situations, Young (1999) proposed the following empirical relationships between nondimensional peak frequency and fetch:
e5
During the land breeze of the morning of 20 February, we calculated values of fetch (upwind distance to the shoreline) and peak frequency at points B, C, and D (offshore from A in deeper water; see Fig. 1) from SWAN output. Figure 7 shows Eq. (5) [with accuracy limits given by Young (1999), reflecting the variation among the various experimental datasets used], along with the nondimensional fetch and peak frequency from SWAN. The model output during the land-breeze phase is seen to be consistent with fetch-limited wave growth.
Fig. 7.
Fig. 7.

Nondimensional peak frequency ν against nondimensional fetch χ at three offshore locations at hourly intervals between 0400 and 0700 LT 20 Feb. Also shown is the relationship according to Eq. (5) with accuracy limits from Young (1999).

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

Observations of duration-limited wave growth are not as common as fetch-limited cases (Hwang and Wang 2004; Young 1999). From the onset of the sea breeze at 1300 LT on 18 February, the regime clearly changes. It will now be examined if the new wave growth regime becomes duration limited. Following Holthuijsen [2007, Eq. (6.3.2)], we calculate the equivalent fetch:
e6
where cg,peak is the group velocity at the peak frequency. Table 1 shows values of equivalent fetch calculated from the peak frequencies of the AWAC and SWAN spectra on the afternoon of 18 February. Note that the modeled winds put the onset of the sea breeze approximately an hour earlier than the observed winds. We have used the model winds over the medium grid to estimate the actual fetch during the sea-breeze cycle. The wind field at 1400, 1600 and 1800 LT on February 18 are shown in Fig. 8 with range rings centered on the AWAC. We estimate the actual fetch as the distance upwind from the AWAC to the point where the offshore wind direction is significantly different from that at the AWAC. Shown in Table 1 are our somewhat subjective estimates of the actual fetch, which is over 20 km within an hour of the onset of the sea breeze, then increases to over 80 km at the peak of the sea breeze. In each case, the actual fetch is much longer than the equivalent fetch throughout the afternoon, with the values converging toward the evening. Hence, according to Holthuijsen (2007), the duration is the limiting factor on wave growth during the sea breeze.
Table 1.

Duration (hours since onset of sea breeze at 1300 LT 18 Feb) and fetch values on the afternoon of 18 Feb. Equivalent fetch Feq calculated from Eq. (6) using the AWAC and the SWAN spectra. Actual fetch values Factual are estimated from the RAMS winds fields in Fig. 8.

Table 1.
Fig. 8.
Fig. 8.

Wind field from RAMS (medium grid) for the afternoon of 18 Feb from the onset of the sea breeze at (a) 1400 LT then at (b) 1600 and (c) 1800 LT. Also shown are concentric lines of 20-, 40-, and 80-km radii, indicating distance from the AWAC site at Secret Harbour.

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

Figure 9 shows the modeled two-dimensional wave energy spectrum at the AWAC location at 1800 LT on 19 February, when the wind is coming from the southwest at 9.1 m s−1, the time of the maximum wind-sea peak. Also shown are the most significant source/sink terms in Eq. (1). That is, wind input Sin, dissipation through whitecapping Sds, and quadruplet interactions Snl, along with the local wind vector. One can see that the region of high wind input is also where energy is lost to whitecapping and spread through wave–wave interactions (Figs. 9c,d).

Fig. 9.
Fig. 9.

At location A, two-dimensional (frequency, direction) spectra of (a) log(N) wave action density, (b) Sin and wind speed U10, (c) Sds, and (d) Snl. The local wind is indicated by the white arrow.

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

The generated wind sea also has significant energy in spectral components traveling in directions away from the peak wind-sea direction (Fig. 9a). This is mainly because of quadruplet interactions (Fig. 9d) enabling energy in various wave bins to interact and add energy to wave bins at other frequencies and directions. Also, the Sin term can inject energy into direction bins up to ±90° because of its cosine power relation.

Over the next 16 h, the wind then veers to the south and intensifies into a land breeze. Figure 10 shows the two-dimensional energy and source/sink terms every 4 h, from 2000 LT on 19 February until 1200 LT on 20 February. The rate of turning of the wind is slow enough to allow the wind-sea peak to be spread in the direction of turning without creating a separate high-frequency peak, as shown in Young (1999) and Young et al. (1987). While the wind direction is oblique to the shoreline, the wind-sea response to the slanting fetch shows preferential wind-sea growth in the longshore direction, with higher-frequency waves more aligned with the wind direction, which is consistent with Ardhuin et al. (2007).

Fig. 10.
Fig. 10.

2D spectra of (left) log(N), (middle) Sin, and (right) Snl every 4 h from 2000 LT 19 Feb to 1200 LT 20 Feb.

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

At 1200 LT on 20 February, the source/sink terms have very low values. Apparent in the wave energy at this time (Fig. 10, last row) are three localized peaks in frequency-direction space. The main peak in wave energy is propagating in the opposite direction of the wind and represents remnants of the previous swell and wind sea. Two other peaks are evident, both at higher frequencies (0.6–0.8 Hz), but traveling in opposite directions that align with the angle of the shoreline nearby. The presence of these latter peaks create the apparent persistence of high energy at high frequencies and decay of wave energy at midfrequencies (0.4–0.5 Hz), as seen in the observed and modeled one-dimensional spectra presented earlier (Figs. 3, 4).

b. Longshore modes

The shore-parallel peaks of high-frequency wave energy are advected into this location at a rate following the local group velocity from source regions in both longshore directions. The peak that is traveling in a northward direction is wind-sea generated, to the south, during the turning wind field as seen in Fig. 9, including slanting-fetch growth when the wind is coming from the southeast. To investigate the source of the southward-propagating wave energy, the model spectra at site E (see Fig. 1), 1000 m north of site A and the same distance from shore, are now examined (Fig. 11). Unlike at site A, at 1200 LT on 20 February, at site E there is input from the wind as shown in the Sin term in Fig. 11b, causing wind-sea growth at E with wave energy spread between ±90° of the wind direction.

Fig. 11.
Fig. 11.

As in Fig. 9, but at site E at 1200 LT 20 Feb.

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

Comparing the wind input at these two locations at this time, there are two main differences between conditions at the A and E locations, water depth (8.9 and 6.3 m, respectively) and wind speed (2.9 and 3.3 m s−1, respectively). Equation (3) shows that the coefficient of exponential growth has a depth dependence through the phase velocity. At both these depths the phase speed is for deep-water waves , for frequencies above 0.4 Hz. However, the wave growth at both locations is fetch-limited starting from the shoreline. Hence, we expect the shallower bathymetry between E and the shoreline (see Fig. 1) would contribute to more wave growth. The land breeze is stronger at E relative to A, from 5% stronger at 0500 LT to 15% stronger at 1200 LT on February 20, and this difference is also seen in the land breeze of the previous day. Equation (3) shows that the growth term has a linear dependence on . Thus, the difference in wind speeds at the two locations also means there is greater wave growth term at site E. The terms Sin and Snl (Figs. 11b,d) spread the wind-sea energy in direction space. Hence, some high-frequency energy from the vicinity of location E propagates southward to location A, appearing as the southward-traveling peak in Fig. 10 (last row), a result typical for slanting-fetch conditions.

The notion that it is the presence of the strong land breeze, during the night and morning of 20 February, that causes the two high-frequency peaks in longshore directions as seen in Fig. 5a was tested by running SWAN over 19–20 February, turning off all wind at 2000 LT on 19 February. The resulting energy spectrum at 1100 LT on 20 February is shown in Fig. 12.

Fig. 12.
Fig. 12.

Two-dimensional energy spectrum at location A at 1200 LT 20 Feb for the SWAN run where wind is turned off at 2000 LT 19 Feb.

Citation: Journal of Physical Oceanography 44, 12; 10.1175/JPO-D-13-0205.1

Comparing this plot to the energy spectrum at the same time and location in SWAN when the land breeze has been blowing (Fig. 9), we can see that, in the absence of the veering southerly wind followed by an easterly land breeze, the two localized high-frequency peaks traveling in longshore directions are not present. The wind sea at higher frequencies propagating from the southwest from the previous sea breeze shows the high-frequency tail of a typical JONSWAP spectrum, which is quite different from the frequency dependence when the land breeze is present.

In this case of sea-breeze conditions turning and becoming land-breeze conditions over several hours, the wind-sea peak is able to smoothly follow the wind direction. A consequence of this is that the quadruplet interactions, which act to remove energy from the wind-sea peak and add energy to high frequencies, tend to add energy to the waves being generated by the turning wind. Hence, the whitecapping dissipation acting on the older wind sea has more effect when the wind turns. If the wind is turned off after the sea breeze duration, the wind sea at high frequencies evolves, keeping a stable JONSWAP-like shape due to the balance between whitecapping and quadruplet interactions. In the absence of wind forcing, the energy at high frequencies is maintained by the balance between loss due to whitecapping and gain by quadruplet interactions or the shape-stabilizing effect of the nonlinear term (Young 1999). It is possible that during the turning wind phase, the nonstationary computation in SWAN had not sufficiently converged at location A near the coast. SWAN was rerun with the convergence criteria halved, and the solution was seen to converge at the coastal locations to the same state as in the previous runs.

When a turning or opposing wind occurs, the quadruplet interactions act to distribute the energy in directional space, and the gain of energy at high frequencies is transferred to a different part of the directional spectrum. Bottema and van Vledder (2008) found that compared to more accurate schemes, the DIA scheme in SWAN can both overestimate the spreading of wave energy in directional space and underestimate the amount of wave steering under slanting-fetch conditions. A more accurate calculation of the quadruplet interactions could improve the amount of directional spreading and wave steering and hence the magnitude of the longshore-propagating peaks. Indeed, it is unfortunate that the AWAC directional spectrum does not measure frequencies above 0.5 Hz so that the size of the peaks could be validated. However, the AWAC 1D spectra do show similar energy levels as SWAN simulates, around the frequency range and timing of the longshore peaks.

6. Conclusions

Observations of wave growth and decay during a sea-breeze/land-breeze cycle show a complex evolution. Simulations using SWAN have shown a rich interplay of source and sink terms in the spectral wave action balance equation. The default Snyder wind input scheme produces energy levels at midfrequencies more similar to the observations than the Janssen scheme. There was no obvious reason why the schemes produced the different responses; however, this deserves further investigation. For whitecapping, the default Komen scheme overestimates wave energy at high frequencies (f > 0.5 Hz) consistent with previous studies showing that the Ko scheme produces reduced dissipation at high frequencies in the presence of swell (van der Westhuysen et al. 2007; Rogers et al. 2003). The Ko scheme (and other schemes available in SWAN that were evaluated but not presented here) employs measures of the mean spectral wavenumber and steepness to calculate the whitecapping dissipation at a given frequency. These mean quantities can be influenced by the presence of a swell peak at low frequencies, when estimating the whitecapping dissipation at high frequencies. The AB scheme uses a saturation-based expression to calculate the whitecapping dissipation that essentially computes whitecapping dissipation locally in wavenumber space (Babanin and van der Westhuysen 2008; van der Westhuysen et al. 2007; Alves and Banner 2003). This scheme was found to produce wave energy frequency dependence most similar to the observations.

Here, we have presented illustrative examples of fetch- and duration-limited situations, both from observations and modeling. The nondimensional peak frequency for growth during sea-breeze and land-breeze cases showed how these cases could be classified into duration-limited and fetch-limited, respectively.

At the onset of the sea breeze the energy and peak frequency of the local wind sea increased and decreased respectively with a high-frequency tail between f5 and f3, corresponding to the deep- and shallow-water limits. During this phase, wind-sea growth is duration limited, approaching fetch-limited conditions near the peak of the sea breeze. Apart from Young (1999), Hwang and Wang (2004), and Hwang et al. (2011), there has been little coverage of wave growth under duration-limited conditions. A theoretical treatment of the relationship between fetch and duration (Fontaine 2013) puts the empirical power laws in an analytical context. A possibility for further analysis of our observed and modeled spectra is to apply these new approaches.

During the night a land breeze blows directly offshore, and wind-sea growth was shown to be fetch limited at SWAN locations farther offshore than the AWAC location. By examining the two-dimensional spectra at nearby locations in SWAN, it was found that longshore variations in wind speed can produce significant differences in local wind-sea growth. At the AWAC location, there was no wind-sea growth during the land breeze; however, wave energy is present at high frequencies in both longshore directions. The northward-traveling high-frequency peak arises because of slanting-fetch, while the wind was turning through the south. The southward-traveling peak arises from nearby locations where the wind was slightly stronger and the water depth shallower up fetch, allowing wind-sea growth that spread in directional space. The traveling peaks simulated by SWAN may be overestimated by the DIA approximation for the quadruplet interactions; however, the AWAC 1D spectra show similar energy levels at these frequencies and times.

Previous observation or modeling studies have not shown this longshore propagation of wave energy, in both directions at high frequencies, in the presence of an offshore wind at fetch-limited locations. This study shows that SWAN is quite capable of capturing the behavior of several peaks of wave energy with varied peak frequencies and directions, arising from quite varied histories.

Acknowledgments

Financial support for this research is provided by the CSIRO, the Bureau of Meteorology, and the Royal Australian Navy as part of the Bluelink project. Chaojiao Sun provided many helpful suggestions. The authors also acknowledge the constructive and insightful comments from two anonymous reviewers that lead to significant improvements in the final paper.

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  • van der Westhuysen, A. J., , M. Zijlema, , and J. A. Battjes, 2007: Nonlinear saturation-based whitecapping dissipation in SWAN for deep and shallow water. Coastal Eng., 54, 151170, doi:10.1016/j.coastaleng.2006.08.006.

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  • Young, I. R., 1999: Wind Generated Ocean Waves. Elsevier Ocean Engineering Book Series, Vol. 2, Elsevier, 288 pp.

  • Young, I. R., , and L. A. Verhagen, 1996a: The growth of fetch-limited waves in water of finite-depth. Part 1. Total energy and peak frequency. Coastal Eng., 29, 4778, doi:10.1016/S0378-3839(96)00006-3.

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  • Young, I. R., , and L. A. Verhagen, 1996b: The growth of fetch-limited waves in water of finite-depth. Part 2. Spectral evolution. Coastal Eng., 29, 7999, doi:10.1016/S0378-3839(96)00007-5.

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  • Young, I. R., , and L. A. Verhagen, 1996c: The growth of fetch-limited waves in water of finite-depth. Part 3. Directional spectra. Coastal Eng., 29, 101122, doi:10.1016/S0378-3839(96)00026-9.

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  • Young, I. R., , and A. van Agthoven, 1997: The response of waves to a sudden change in wind speed. Nonlinear Ocean Waves, W. Perrie, Ed., Advances in Fluid Mechanics Series, Vol. 17, WIT Press, 133–162.

  • Young, I. R., , S. Hasselmann, , and K. Hasselmann, 1987: Computations of the response of a wave spectrum to a sudden change in the wind direction. J. Phys. Oceanogr., 17, 13171338, doi:10.1175/1520-0485(1987)017<1317:COTROA>2.0.CO;2.

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